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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, general topology (or point set topology) is the branch of
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology,
geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...
, and
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
. The fundamental concepts in point-set topology are ''continuity'', ''compactness'', and ''connectedness'': *
Continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s, intuitively, take nearby points to nearby points. *
Compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
s are those that can be covered by finitely many sets of arbitrarily small size. * Connected sets are sets that cannot be divided into two pieces that are far apart. The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a ''topology''. A set with a topology is called a ''
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''. ''
Metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s'' are an important class of topological spaces where a real, non-negative distance, also called a '' metric'', can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.


History

General topology grew out of a number of areas, most importantly the following: *the detailed study of subsets of the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
(once known as the ''topology of point sets''; this usage is now obsolete) *the introduction of the
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
concept *the study of
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s, especially normed linear spaces, in the early days of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
. General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics.


A topology on a set

Let ''X'' be a set and let ''τ'' be a
family Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...
of
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s of ''X''. Then ''τ'' is called a ''topology on X'' if: # Both the
empty set In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
and ''X'' are elements of ''τ'' # Any union of elements of ''τ'' is an element of ''τ'' # Any
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of finitely many elements of ''τ'' is an element of ''τ'' If ''τ'' is a topology on ''X'', then the pair (''X'', ''τ'') is called a ''topological space''. The notation ''Xτ'' may be used to denote a set ''X'' endowed with the particular topology ''τ''. The members of ''τ'' are called ''
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s'' in ''X''. A subset of ''X'' is said to be closed if its complement is in ''τ'' (i.e., its complement is open). A subset of ''X'' may be open, closed, both ( clopen set), or neither. The empty set and ''X'' itself are always both closed and open.


Basis for a topology

A base (or basis) ''B'' for a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'' with
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
''T'' is a collection of
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s in ''T'' such that every open set in ''T'' can be written as a union of elements of ''B''. We say that the base ''generates'' the topology ''T''. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because many topologies are most easily defined in terms of a base that generates them.


Subspace and quotient

Every subset of a topological space can be given the
subspace topology In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
in which the open sets are the intersections of the open sets of the larger space with the subset. For any
indexed family In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers, is a collection of real numbers, wher ...
of topological spaces, the product can be given the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
, which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space. A quotient space is defined as follows: if ''X'' is a topological space and ''Y'' is a set, and if ''f'' : ''X''→ ''Y'' is a
surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
function, then the quotient topology on ''Y'' is the collection of subsets of ''Y'' that have open inverse images under ''f''. In other words, the quotient topology is the finest topology on ''Y'' for which ''f'' is continuous. A common example of a quotient topology is when an equivalence relation is defined on the topological space ''X''. The map ''f'' is then the natural projection onto the set of equivalence classes.


Examples of topological spaces

A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space.


Discrete and trivial topologies

Any set can be given the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
, in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given the
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
(also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must be
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
s where limit points are unique.


Cofinite and cocountable topologies

Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T1 topology on any infinite set. Any set can be given the
cocountable topology The cocountable topology, also known as the countable complement topology, is a topology that can be defined on any infinite set X. In this topology, a set is open if its complement in X is either countable or equal to the entire set. Equivalen ...
, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations.


Topologies on the real and complex numbers

There are many ways to define a topology on R, the set of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. The standard topology on R is generated by the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
s R''n'' can be given a topology. In the usual topology on R''n'' the basic open sets are the open balls. Similarly, C, the set of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s, and C''n'' have a standard topology in which the basic open sets are open balls. The real line can also be given the
lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topology defined on \mathbb, the set of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of in ...
. Here, the basic open sets are the half open intervals [''a'', ''b''). This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.


The metric topology

Every
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is the same for all norms.


Further examples

* There exist numerous topologies on any given finite set. Such spaces are called
finite topological space In mathematics, a finite topological space is a topological space for which the underlying set (mathematics), point set is finite set, finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are ...
s. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general. * Every
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
has a natural topology, since it is locally Euclidean. Similarly, every simplex and every
simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
inherits a natural topology from Rn. * The Zariski topology is defined algebraically on the spectrum of a ring or an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
. On R''n'' or C''n'', the closed sets of the Zariski topology are the solution sets of systems of
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
equations. * A linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices and edges. * Many sets of
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function. * Any local field has a topology native to it, and this can be extended to vector spaces over that field. * The
Sierpiński space In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński. The ...
is the simplest non-discrete topological space. It has important relations to the theory of computation and semantics. * If Γ is an
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
, then the set Γ = , Γ) may be endowed with the order topology generated by the intervals (''a'', ''b''), [0, ''b'') and (''a'', Γ) where ''a'' and ''b'' are elements of Γ.


Continuous functions

Continuity is expressed in terms of neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
s: is continuous at some point if and only if for any neighborhood of , there is a neighborhood of such that . Intuitively, continuity means no matter how "small" becomes, there is always a containing that maps inside and whose image under contains . This is equivalent to the condition that the preimages of the open (closed) sets in are open (closed) in . In metric spaces, this definition is equivalent to the ε–δ-definition that is often used in analysis. An extreme example: if a set is given the discrete topology">epsilon-delta definition">ε–δ-definition that is often used in analysis. An extreme example: if a set is given the discrete topology, all functions :f\colon X \rightarrow T to any topological space are continuous. On the other hand, if is equipped with the indiscrete topology and the space set is at least T0, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.


Alternative definitions

Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.


Neighborhood definition

Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms of
neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
s: ''f'' is continuous at some point ''x'' ∈ ''X'' if and only if for any neighborhood ''V'' of ''f''(''x''), there is a neighborhood ''U'' of ''x'' such that ''f''(''U'') ⊆ ''V''. Intuitively, continuity means no matter how "small" ''V'' becomes, there is always a ''U'' containing ''x'' that maps inside ''V''. If ''X'' and ''Y'' are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at ''x'' and ''f''(''x'') instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance. Note, however, that if the target space is Hausdorff, it is still true that ''f'' is continuous at ''a'' if and only if the limit of ''f'' as ''x'' approaches ''a'' is ''f''(''a''). At an isolated point, every function is continuous.


Sequences and nets

In several contexts, the topology of a space is conveniently specified in terms of limit points. In many instances, this is accomplished by specifying when a point is the
limit of a sequence As the positive integer n becomes larger and larger, the value n\times \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n \times \sin\left(\tfrac1\right) equals 1." In mathematics, the li ...
, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a preordered set in which every finite subset has an upper bound. In other words, it is a non-empty preordered set A such that for any a and b in A there exists c in A wit ...
, known as nets. A function is continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition. In detail, a function ''f'': ''X'' → ''Y'' is sequentially continuous if whenever a sequence (''x''''n'') in ''X'' converges to a limit ''x'', the sequence (''f''(''x''''n'')) converges to ''f''(''x''). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If ''X'' is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if ''X'' is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.


Closure operator definition

Instead of specifying the open subsets of a topological space, the topology can also be determined by a closure operator (denoted cl), which assigns to any subset ''A'' ⊆ ''X'' its closure, or an interior operator (denoted int), which assigns to any subset ''A'' of ''X'' its interior. In these terms, a function :f\colon (X,\mathrm) \to (X' ,\mathrm')\, between topological spaces is continuous in the sense above if and only if for all subsets ''A'' of ''X'' :f(\mathrm(A)) \subseteq \mathrm'(f(A)). That is to say, given any element ''x'' of ''X'' that is in the closure of any subset ''A'', ''f''(''x'') belongs to the closure of ''f''(''A''). This is equivalent to the requirement that for all subsets ''A''' of ''X''' :f^(\mathrm'(A')) \supseteq \mathrm(f^(A')). Moreover, :f\colon (X,\mathrm) \to (X' ,\mathrm') \, is continuous if and only if :f^(\mathrm'(A)) \subseteq \mathrm(f^(A)) for any subset ''A'' of ''X''.


Properties

If ''f'': ''X'' → ''Y'' and ''g'': ''Y'' → ''Z'' are continuous, then so is the composition ''g'' ∘ ''f'': ''X'' → ''Z''. If ''f'': ''X'' → ''Y'' is continuous and * ''X'' is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
, then ''f''(''X'') is compact. * ''X'' is connected, then ''f''(''X'') is connected. * ''X'' is path-connected, then ''f''(''X'') is path-connected. * ''X'' is Lindelöf, then ''f''(''X'') is Lindelöf. * ''X'' is separable, then ''f''(''X'') is separable. The possible topologies on a fixed set ''X'' are partially ordered: a topology τ1 is said to be coarser than another topology τ2 (notation: τ1 ⊆ τ2) if every open subset with respect to τ1 is also open with respect to τ2. Then, the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
:idX: (''X'', τ2) → (''X'', τ1) is continuous if and only if τ1 ⊆ τ2 (see also
comparison of topologies In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as t ...
). More generally, a continuous function :(X, \tau_X) \rightarrow (Y, \tau_Y) stays continuous if the topology τ''Y'' is replaced by a coarser topology and/or τ''X'' is replaced by a finer topology.


Homeomorphisms

Symmetric to the concept of a continuous map is an
open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...
, for which ''images'' of open sets are open. In fact, if an open map ''f'' has an
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
, that inverse is continuous, and if a continuous map ''g'' has an inverse, that inverse is open. Given a
bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
function ''f'' between two topological spaces, the inverse function ''f''−1 need not be continuous. A bijective continuous function with continuous inverse function is called a ''
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
''. If a continuous bijection has as its domain a
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
and its
codomain In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term '' range'' is sometimes ambiguously used to ...
is Hausdorff, then it is a homeomorphism.


Defining topologies via continuous functions

Given a function :f\colon X \rightarrow S, \, where ''X'' is a topological space and ''S'' is a set (without a specified topology), the final topology on ''S'' is defined by letting the open sets of ''S'' be those subsets ''A'' of ''S'' for which ''f''−1(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on ''S''. Thus the final topology can be characterized as the finest topology on ''S'' that makes ''f'' continuous. If ''f'' is
surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
, this topology is canonically identified with the quotient topology under the equivalence relation defined by ''f''. Dually, for a function ''f'' from a set ''S'' to a topological space ''X'', the initial topology on ''S'' has a basis of open sets given by those sets of the form ''f''−1(''U'') where ''U'' is open in ''X'' . If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on ''S''. Thus the initial topology can be characterized as the coarsest topology on ''S'' that makes ''f'' continuous. If ''f'' is injective, this topology is canonically identified with the
subspace topology In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
of ''S'', viewed as a subset of ''X''. A topology on a set ''S'' is uniquely determined by the class of all continuous functions S \rightarrow X into all topological spaces ''X''. Dually, a similar idea can be applied to maps X \rightarrow S.


Compact sets

Formally, a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'' is called ''compact'' if each of its
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\su ...
s has a finite subcover. Otherwise it is called ''non-compact''. Explicitly, this means that for every arbitrary collection :\_ of open subsets of such that :X = \bigcup_ U_\alpha, there is a finite subset of such that :X = \bigcup_ U_i. Some branches of mathematics such as
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, typically influenced by the French school of Bourbaki, use the term ''quasi-compact'' for the general notion, and reserve the term ''compact'' for topological spaces that are both Hausdorff and ''quasi-compact''. A compact set is sometimes referred to as a ''compactum'', plural ''compacta''. Every closed interval in R of finite length is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
. More is true: In Rn, a set is compact
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is closed and bounded. (See Heine–Borel theorem). Every continuous image of a compact space is compact. A compact subset of a Hausdorff space is closed. Every continuous
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
from a compact space to a Hausdorff space is necessarily a
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
. Every
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of points in a compact metric space has a convergent subsequence. Every compact finite-dimensional
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
can be embedded in some Euclidean space Rn.


Connected sets

A
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'' is said to be disconnected if it is the union of two disjoint nonempty
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s. Otherwise, ''X'' is said to be connected. A
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of a topological space is said to be connected if it is connected under its
subspace topology In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
. Some authors exclude the
empty set In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
(with its unique topology) as a connected space, but this article does not follow that practice. For a topological space ''X'' the following conditions are equivalent: #''X'' is connected. #''X'' cannot be divided into two disjoint nonempty
closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
s. #The only subsets of ''X'' that are both open and closed ( clopen sets) are ''X'' and the empty set. #The only subsets of ''X'' with empty boundary are ''X'' and the empty set. #''X'' cannot be written as the union of two nonempty separated sets. #The only continuous functions from ''X'' to , the two-point space endowed with the discrete topology, are constant. Every interval in R is connected. The continuous image of a connected space is connected.


Connected components

The maximal connected subsets (ordered by inclusion) of a nonempty topological space are called the connected components of the space. The components of any topological space ''X'' form a partition of ''X'': they are disjoint, nonempty, and their union is the whole space. Every component is a closed subset of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s are the one-point sets, which are not open. Let \Gamma_x be the connected component of ''x'' in a topological space ''X'', and \Gamma_x' be the intersection of all open-closed sets containing ''x'' (called quasi-component of ''x''.) Then \Gamma_x \subset \Gamma'_x where the equality holds if ''X'' is compact Hausdorff or locally connected.


Disconnected spaces

A space in which all components are one-point sets is called totally disconnected. Related to this property, a space ''X'' is called totally separated if, for any two distinct elements ''x'' and ''y'' of ''X'', there exist disjoint open neighborhoods ''U'' of ''x'' and ''V'' of ''y'' such that ''X'' is the union of ''U'' and ''V''. Clearly any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbers Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.


Path-connected sets

A '' path'' from a point ''x'' to a point ''y'' in a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'' is a
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
''f'' from the unit interval ,1to ''X'' with ''f''(0) = ''x'' and ''f''(1) = ''y''. A '' path-component'' of ''X'' is an equivalence class of ''X'' under the equivalence relation, which makes ''x'' equivalent to ''y'' if there is a path from ''x'' to ''y''. The space ''X'' is said to be '' path-connected'' (or ''pathwise connected'' or ''0-connected'') if there is at most one path-component; that is, if there is a path joining any two points in ''X''. Again, many authors exclude the empty space. Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended long line ''L''* and the '' topologist's sine curve''. However, subsets of the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
R are connected
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
they are path-connected; these subsets are the intervals of R. Also, open subsets of R''n'' or C''n'' are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for
finite topological space In mathematics, a finite topological space is a topological space for which the underlying set (mathematics), point set is finite set, finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are ...
s.


Products of spaces

Given ''X'' such that :X := \prod_ X_i, is the Cartesian product of the topological spaces ''Xi'', indexed by i \in I, and the canonical projections ''pi'' : ''X'' → ''Xi'', the product topology on ''X'' is defined as the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections ''pi'' are continuous. The product topology is sometimes called the Tychonoff topology. The open sets in the product topology are unions (finite or infinite) of sets of the form \prod_ U_i, where each ''Ui'' is open in ''Xi'' and ''U''''i'' ≠ ''X''''i'' only finitely many times. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the ''Xi'' gives a basis for the product \prod_ X_i. The product topology on ''X'' is the topology generated by sets of the form ''pi''−1(''U''), where ''i'' is in ''I '' and ''U'' is an open subset of ''Xi''. In other words, the sets form a subbase for the topology on ''X''. A
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of ''X'' is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form ''pi''−1(''U''). The ''pi''−1(''U'') are sometimes called open cylinders, and their intersections are cylinder sets. In general, the product of the topologies of each ''Xi'' forms a basis for what is called the box topology on ''X''. In general, the box topology is finer than the product topology, but for finite products they coincide. Related to compactness is
Tychonoff's theorem In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is tra ...
: the (arbitrary) product of compact spaces is compact.


Separation axioms

Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms; for example, the meanings of "normal" and "T4" are sometimes interchanged, similarly "regular" and "T3", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous. Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles. In all of the following definitions, ''X'' is again a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
. * ''X'' is '' T0'', or ''Kolmogorov'', if any two distinct points in ''X'' are topologically distinguishable. (It is a common theme among the separation axioms to have one version of an axiom that requires T0 and one version that doesn't.) * ''X'' is '' T1'', or ''accessible'' or ''Fréchet'', if any two distinct points in ''X'' are separated. Thus, ''X'' is T1 if and only if it is both T0 and R0. (Though you may say such things as ''T1 space'', ''Fréchet topology'', and ''Suppose that the topological space ''X'' is Fréchet'', avoid saying ''Fréchet space'' in this context, since there is another entirely different notion of Fréchet space in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
.) * ''X'' is '' Hausdorff'', or ''T2'' or ''separated'', if any two distinct points in ''X'' are separated by neighbourhoods. Thus, ''X'' is Hausdorff if and only if it is both T0 and R1. A Hausdorff space must also be T1. * ''X'' is '' T2½'', or ''Urysohn'', if any two distinct points in ''X'' are separated by closed neighbourhoods. A T2½ space must also be Hausdorff. * ''X'' is '' regular'', or ''T3'', if it is T0 and if given any point ''x'' and closed set ''F'' in ''X'' such that ''x'' does not belong to ''F'', they are separated by neighbourhoods. (In fact, in a regular space, any such ''x'' and ''F'' is also separated by closed neighbourhoods.) * ''X'' is '' Tychonoff'', or ''T3½'', ''completely T3'', or ''completely regular'', if it is T0 and if f, given any point ''x'' and closed set ''F'' in ''X'' such that ''x'' does not belong to ''F'', they are separated by a continuous function. * ''X'' is '' normal'', or ''T4'', if it is Hausdorff and if any two disjoint closed subsets of ''X'' are separated by neighbourhoods. (In fact, a space is normal if and only if any two disjoint closed sets can be separated by a continuous function; this is Urysohn's lemma.) * ''X'' is '' completely normal'', or ''T5'' or ''completely T4'', if it is T1 and if any two separated sets are separated by neighbourhoods. A completely normal space must also be normal. * ''X'' is '' perfectly normal'', or ''T6'' or ''perfectly T4'', if it is T1 and if any two disjoint closed sets are precisely separated by a continuous function. A perfectly normal Hausdorff space must also be completely normal Hausdorff. The
Tietze extension theorem In topology, the Tietze extension theorem (also known as the Tietze– Urysohn– Brouwer extension theorem or Urysohn-Brouwer lemma) states that any real-valued, continuous function on a closed subset of a normal topological space In mathe ...
: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.


Countability axioms

An axiom of countability is a
property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, re ...
of certain
mathematical object A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...
s (usually in a category) that requires the existence of a
countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
with certain properties, while without it such sets might not exist. Important countability axioms for
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s: * sequential space: a set is open if every
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
convergent to a point in the set is eventually in the set * first-countable space: every point has a countable neighbourhood basis (local base) * second-countable space: the topology has a countable base * separable space: there exists a countable dense subspace * Lindelöf space: every
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\su ...
has a countable subcover *
σ-compact space In mathematics, a topological space is said to be ''σ''-compact if it is the union of countably many compact subspaces. A space is said to be ''σ''-locally compact if it is both ''σ''-compact and (weakly) locally compact. That terminology ...
: there exists a countable cover by compact spaces Relations: *Every first countable space is sequential. *Every second-countable space is first-countable, separable, and Lindelöf. *Every σ-compact space is Lindelöf. *A
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
is first-countable. *For metric spaces second-countability, separability, and the Lindelöf property are all equivalent.


Metric spaces

A metric space is an
ordered pair In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...
(M,d) where M is a set and d is a metric on M, i.e., a function :d \colon M \times M \rightarrow \mathbb such that for any x, y, z \in M, the following holds: # d(x,y) \ge 0     (''non-negative''), # d(x,y) = 0\,
iff In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both ...
x = y\,     ('' identity of indiscernibles''), # d(x,y) = d(y,x)\,     (''symmetry'') and # d(x,z) \le d(x,y) + d(y,z)     (''
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...
'') . The function d is also called ''distance function'' or simply ''distance''. Often, d is omitted and one just writes M for a metric space if it is clear from the context what metric is used. Every
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
is
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...
and Hausdorff, and thus normal. The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.


Baire category theorem

The Baire category theorem says: If ''X'' is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty. Any open subspace of a Baire space is itself a Baire space.


Main areas of research


Continuum theory

A continuum (pl ''continua'') is a nonempty
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
connected
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, or less frequently, a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
connected
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
. Continuum theory is the branch of topology devoted to the study of continua. These objects arise frequently in nearly all areas of topology and
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
, and their properties are strong enough to yield many 'geometric' features.


Dynamical systems

Topological dynamics concerns the behavior of a space and its subspaces over time when subjected to continuous change. Many examples with applications to physics and other areas of math include
fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
, billiards and flows on manifolds. The topological characteristics of
fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
s in fractal geometry, of
Julia set In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of ...
s and the Mandelbrot set arising in complex dynamics, and of attractors in differential equations are often critical to understanding these systems.


Pointless topology

Pointless topology (also called point-free or pointfree topology) is an approach to
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
that avoids mentioning points. The name 'pointless topology' is due to
John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
.Garrett Birkhoff, ''VON NEUMANN AND LATTICE THEORY'', ''John Von Neumann 1903-1957'', J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 50-5 The ideas of pointless topology are closely related to mereotopologies, in which regions (sets) are treated as foundational without explicit reference to underlying point sets.


Dimension theory

Dimension theory is a branch of general topology dealing with dimensional invariants of
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s.


Topological algebras

A topological algebra ''A'' over a topological field K is a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
together with a continuous multiplication :\cdot :A\times A \longrightarrow A :(a,b)\longmapsto a\cdot b that makes it an
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
over K. A unital
associative In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
topological algebra is a topological ring. The term was coined by David van Dantzig; it appears in the title of his doctoral dissertation (1931).


Metrizability theory

In
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
and related areas of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a metrizable space is a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
that is
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
. That is, a topological space (X,\tau) is said to be metrizable if there is a metric :d\colon X \times X \to ,\infty) such that the topology induced by ''d'' is \tau. Metrization theorems are theorems that give sufficient condition">theorem">,\infty) such that the topology induced by ''d'' is \tau. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.


Set-theoretic topology

Set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC). A famous problem is Moore space (topology)#Normal Moore space conjecture, the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.


See also

* List of examples in general topology * Glossary of general topology for detailed definitions * List of general topology topics for related articles *
Category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...


References


Further reading

Some standard books on general topology include: * Bourbaki, Topologie Générale (General Topology), . * * Stephen Willard, General Topology, . * James Munkres, Topology, . * George F. Simmons, Introduction to Topology and Modern Analysis, . * Paul L. Shick, Topology: Point-Set and Geometric, . * Ryszard Engelking, General Topology, . * * O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev
Elementary Topology: Textbook in Problems
. The
arXiv arXiv (pronounced as "archive"—the X represents the Chi (letter), Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not Scholarly pee ...
subject code i
math.GN


External links

* {{Authority control